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Simplifying 4w2 + 12w + -14 = 0 Reorder the terms: -14 + 12w + 4w2 = 0 Solving -14 + 12w + 4w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 6w + 2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 6w + 2w2)' equal to zero and attempt to solve: Simplifying -7 + 6w + 2w2 = 0 Solving -7 + 6w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 3w + w2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 3w + 3.5 + w2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 3w + w2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 3w + w2 = 0 + 3.5 3w + w2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 3w + w2 = 3.5 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 3.5 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 3.5 + 2.25 Combine like terms: 3.5 + 2.25 = 5.75 2.25 + 3w + w2 = 5.75 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (w + 1.5) equal to 2.397915762 and -2.397915762.Subproblem 1
w + 1.5 = 2.397915762 Simplifying w + 1.5 = 2.397915762 Reorder the terms: 1.5 + w = 2.397915762 Solving 1.5 + w = 2.397915762 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 2.397915762 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 2.397915762 + -1.5 w = 2.397915762 + -1.5 Combine like terms: 2.397915762 + -1.5 = 0.897915762 w = 0.897915762 Simplifying w = 0.897915762Subproblem 2
w + 1.5 = -2.397915762 Simplifying w + 1.5 = -2.397915762 Reorder the terms: 1.5 + w = -2.397915762 Solving 1.5 + w = -2.397915762 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -2.397915762 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -2.397915762 + -1.5 w = -2.397915762 + -1.5 Combine like terms: -2.397915762 + -1.5 = -3.897915762 w = -3.897915762 Simplifying w = -3.897915762Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.897915762, -3.897915762}Solution
w = {0.897915762, -3.897915762}
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